Problem: Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?  Express your answer as a common fraction. [asy]
/* AMC8 2004 #21 Problem */
draw(circle((0,0), 2));
draw(circle((6,0), 2));
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
path p=(0,0)--(0,2);
draw(shift(6,0)*p);
draw(shift(6,0)*rotate(120)*p);
draw(shift(6,0)*rotate(240)*p);
label(scale(0.6)*"1", (-1,1));
label(scale(0.6)*"2", (1,1));
label(scale(0.6)*"3", (1,-1));
label(scale(0.6)*"4", (-1,-1));
label(scale(0.6)*"1", (5,.75));
label(scale(0.6)*"2", (7,.75));
label(scale(0.6)*"3", (6,-1));
draw((0,0)--(1.5,.5), EndArrow);
draw(shift(6,0)*rotate(45)*((0,0)--(1.5,.5)), EndArrow);
[/asy]
Explanation: In eight of the twelve outcomes the product is even: $1\times 2$, $2\times
1$, $2\times 2$, $2\times 3$, $3\times 2$, $4\times 1$, $4\times 2$, $4\times 3$. In four of the twelve, the product is odd: $1\times 1$, $1\times 3$, $3\times 1$, $3\times 3$. So the probability that the product is even is $\frac{8}{12}$ or $\boxed{\frac{2}{3}}$.

OR

To get an odd product, the result of both spins must be odd. The probability of odd is $\frac{1}{2}$ on Spinner $A$ and $\frac{2}{3}$ on Spinner $B$. So the probability of an odd product is $\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)=\frac{1}{3}$. The probability of an even product, then, is $1-\frac{1}{3}=\boxed{\frac{2}{3}}$.